Isabel Lipartito Astronomy Department, Smith College, Northampton, MA 01063

Transcription

1 Draft version September 4, 2014 Preprint typeset using L A TEX style emulateapj v. 05/12/14 CALCULATING A STELLAR-SUPERMASSIVE BLACK HOLE MICROLENSING RATE Isabel Lipartito Astronomy Department, Smith College, Northampton, MA and Andrea Ghez, Gunther Witzel, Breann Sitarski, and Samantha Chappell Astronomy Department, University of California, Los Angeles, Los Angeles, CA, Draft version September 4, 2014 ABSTRACT The Galactic center contains a supermassive black hole whose gravitational field tidally affects nearby objects. One observable effect is relativistic stellar microlensing, or the deflection of stellar light from orbiting stars passing in the vicinity of the black hole. This leads to the generation of multiple unresolved stellar images and an overall increase in brightness of the source. Detection and analysis of microlensing events give information regarding the nature of the black hole and the workings of general relativity. We aim to calculate an updated rate of observable microlensing events for K-Band emitting stars present in the innermost few parsecs of the Galactic center, taking into account current data available from the UCLA Galactic Center Group for stellar densities, velocities, and luminosities. Lensing event rates are calculated both for contemporary observational limits, specifically, those for Keck Observatory and its current adaptive optics system, as well as the limits for telescopes of the future (the Thirty Meter Telescope). Furthermore, an effort is made to take into account the duration of observing runs. This calculation was originally completed in 1999 by Alexander and Sternberg; rates generated suggested detectable lensing events are one event per one thousand years, too rare for us to expect to see on observation timescales of years or even decades. However, with the improved observation parameters of adaptive optics, we can expect to detect one lensing event per 5 years (via Keck AO) for unresolved lensing in the case where observations are spaced by several months or years apart. The new generation of telescopes will have even greater detection potential, as projected detection rates are one event every 1-2 years even without continuous observation (time spacing of months or even years). Subject headings: galactic center, microlensing, lensing, black hole, SMBH 1. INTRODUCTION Multiple routes exist by which one can gather information indirectly about the environment of the Galactic center and its workings, governed by gravitational interactions between the GC Supermassive black hole of 4.2 million solar masses, Do et al. (2013). One of these is the detection of SMBH microlensing events. The Galactic center contains a dense population of S stars orbiting the black hole, whose orbital astrometry has previously been used to constrain measurements for the mass and distance of the black hole from Earth, Ghez et al. (2008). We should expect to thus be able to observe lensing events which occur when these stars pass in the vicinity of the black hole. Lensing is a relativistic phenomenon requiring a lens (here, the SMBH) and a source (the orbiting star). Photons from an orbiting star follow geodesics in the warped space-time around the massive black hole. If the source and lens are lined up directly in our line of sight, we should observe an Einstein ring of multiple images of the source, all formed as the photons coming from the source follow different paths in space-time. Indirect alignment of source and lens relative to

2 2 Lipartito et al our line of sight leads to the creation of two or more lensed images. In this situation, however, the effect we most often shall observe is known as microlensing. The Einstein Radius gives the typical distance between gravitationally lensed images, Alexander & Sternberg (1999), 4 G MBH R E = c 2 r D BH r + D BH Where r is the lens source distance Where D BH is the distance to the SMBH (1) but the optical resolving power of the telescope used to observe the Galactic center determines whether or not we are able to resolve the multiple images. The Keck telescope has an angular resolution threshold of 0.06 arcseconds, translating to an lens-source distance of about 6 pc using a distance to the BH of 8.46 kpc. If we limit our obervations of stellar lensing to consist of stars within the inner 0.5 pc radius of the GC, then all lensing we can contemporarily observe is microlensing, where the source and images cannot be separately resolved, and the observational effect of lensing is a brightening of the source during the lensing event, Alexander & Sternberg (1999). In 1999, a microlensing rate for the inner GC was computed by Alexander and Sternberg using contemporary values for black hole mass, distance from Earth, and the stellar population in the GC (stellar densities, velocities, and luminosities). An unfortunately small rate of detectable lensing events (limited by telescope resolution in the pre-ao era) was computed: about 1-3 events per 1000 years, even in the limit of continuous observation. However, times have changed. In 2014, we have bigger telescopes (e.g. the twin Keck 10 m telescopes), adaptive optics, and greater optical resolving power. We are able to see even fainter magnitudes and we have updated parameters for the SMBH mass, distance, and properties for orbiting stars in the GC from the UCLA Galactic Center Group. We thus decided to update the rate calculations of 1999 taking into consideration the new parameters and data on the Galactic center. Our calculations demonstrated we should be seeing around 1 event per 5 years in the inner GC using our current generation of telescopes and 1 event per year in the inner GC with new telescopes such as TMT. 2. CALCULATING A MICROLENSING RATE 2.1. Basic Equation Integral to the generation of a microlensing rate is correctly understanding which parameters need to be part of the rate calculation and how to implement them. The microlensing rate depends both upon parameters regarding the behavior of stars and relativistic lensing in the GC along with a few parameters describing observational limits. The calculable lensing rate depends, basically, on the velocities of orbiting stars (how often are they going to pass near enough the black hole for lensing), the rotational velocity of the galaxy, and the number density of these stars (how many of them are available for lensing). Each of these parameters are functions of distance from Galactic Center. Another factor in this calculation is the brightness of stars: the expected average K- Band magnitude of stars in the region we are observing along with another parameter designating the highest K-Band stellar magnitude a telescope can successfully detect. Another significant parameter determining the rate is the Einstein Radius of a lensing event, which itself depends on the distance between a lens and source. Now comes the question of how to combine these basic parameters and how to deal with the distance consideration. By making the assumption that the SMBH is directly at the Galactic center, then the lenssource distance variable in the Einstein Radius parameter would be simply the line-of-sight distance between the star and the Galactic center. Excluding the luminosity parameter, all parameters now depend on distance from the Galactic center. As we include a larger distance range, more stars and thus potential lensing events will be included in our scope of observation. Moreover, these parameters individually do not depend on each other. Therefore we multiply these parameters together and integrate over the distance from the Galactic center to a specified line-of-sight distance, whose magnitude depends upon what sort of lensing we want to see. Specific to each instrument, a certain lenssource distance limit will enable us to only see microlensing (call this the microlensing limit ); further beyond that point includes lens-source distances and hence lensing events that can be resolved. Below is a basic template for lensing

3 Calculating a Stellar-Supermassive Black Hole Microlensing Rate 3 rate calculations introduced by Alexander and Sternberg. Included is a parameter for the stellar velocity field, stellar number density, and Einstein Radius, all dependent on the distance from the lens. r2 Γ(K 0 ) = 2 u 0 R E v n dr (2) r Observational Parameters The aforementioned parameters, however, only take into consideration the astrophysical parameters at the Galactic center. We need to be concerned not only with the total number of lensing events that are occurring over a certain time period, but how many of those events we are able to detect, as well as the nature of these events. We need to know the threshold brightness of our telescope (how faint a star can we see). u 0 = 10 K 0 K 2.5 (3) Additionally, it is necessary to know the mean impact parameter, u 0 defined as the ratio of the average flux from a star divided by the detection threshold flux, Alexander & Sternberg (1999). (K 0 is the threshold K magnitude corresponding to the treshold flux, K is the average stellar magnitude corresponding to the average stellar flux.) We need to know the lens-source distance limit for microlensing as well. This value can be obtained by calculating the lens-source distance that will generate an Einstein Radius at the optical resolution of the telescope in question; the minimum distance for resolved lensing. Solve for r : arctan[φ] D BH = 4GMBH c 2 r D BH r + D BH (4) 2.3. Rotational Velocity and Time Dependence In adddition to the peculiar motions of stars in the galaxy, r v rotational = log[ 1 pc ] (5) it is important that we take into account the overall rotational velocity of the galaxy. Alexander and Sternberg provides a function to work with this: G[r] = e v 2 rotational2 2 vmax e y2 2 I 0 [y v rotational2 ] dr 0 y 2 y 1 ( ) v 2 max Where I 0 is a 0 th order modified Bessel function (6) (Note: rotational velocity here, v rotational2, is rotational velocity divided by peculiar velocity, to be made unitless for this function) A k = 2.6 Extinction Coefficient (7) = 5 log[d BH ] 5 Distance Modulus (8) 1 u s = 10.4 (23 K A k (9) v max = 2 u s R E v T (10) Another parameter, v max, goes into this function. This is where the time dependence of observations comes in. In the case of searching for lensing events, the yearly lensing rate must depend on the period of time between observing runs, as some lensing events will occur in that time period and will be undetectable to us. The same function that takes care of rotational velocity also includes a parameter for maximal event velocity: basically, Einstein radius divided by the time between observations. Events occurring at a velocity at this value or higher will happen while the telescope is not in use and will not be observed. This function, called G(r) in Alexander s paper, takes in a unitless rotational velocity and maximum velocity and generates a unitless constant to go into the lensing equation, filtering out events unavailable to us and thus accounting for another limit of observation. 3. REPRODUCING ALEXANDER AND STERNBERG, 1999 Putting this all together, we end up with r2 2 u 0 R E [r] v[r] n [r] G[r] dr (11) r 1

4 4 Lipartito et al where v[r] is the 1-D velocity dispersion and n [r] is the stellar number density adapted from Alexander s paper. R E, G[r], and u 0 are as discussed earlier. Taking into account an optical resolution given for the Keck Era of about 120 milliarcseconds, we find that the outer limit for distance from Galactic center for microlensing is 45 pc. Alexander extends the calculation for total lensing (microlensing and resolved) to 300 pc total. Figure 1 depicts the KLF created and used by Alexander, which predicts an average magnitude for GC stars of about Alexander also suggests a threshold magnitude of 17. Using such parameters and values, I worked to recreate the 1999 time-dependent lensing rates generated by Alexander and was successful, within the same order of magnitude. Most importantly, I was able to recreate the significant result, for a threshold magnitude of 17, a total lensing rate of about 3 events per year. Figures 2, 3, and 4 show the reproduced microlensing (45 pc integration limit) and total lensing (300 pc limit) curves, respectively. These curves are dependent on times between observing runs which range from 0.01 years to 10 years. Calculations were done for a magnitude threshold range, although 17 was the accepted threshold at the time. 4. UPDATING THE CALCULATIONS OF ALEXANDER AND STERNBERG, 1999 Despite the rather unfortunate 1999 result of a couple lensing detectable events per millennium, we found it a promising endeavor to recreate the work of Alexander and Sternberg, taking into account updated parameters for black hole properties (mass, distance from Earth), and stellar parameters in the GC: stellar velocities, number density distribution, and luminosities, adapted from data and results from the UCLA Galactic Center research group. Most importantly, observational parameters have changed. Since 1999, we have the advent of adaptive optics for Keck and the prospect of the construction of the Thirty Meter Telescope in the future. There should be a far larger number of detectable lensing events as we can see even fainter stars. Observational instruments considered were Keck and current AO, Keck and Next Generation AO (NGAO), and the TMT. TABLE 1 SMBH and Stellar Number Density Distribution Parameters from Do et al. (2013) Variable Definition Value r b Break Radius for ρ (pc) 1.51 γ Inner Slope of ρ.22 δ Sharpness in ρ transition 6.81 α Outer Slope in ρ 6.31 M BH BH Mass transition ( 10 6 Solar Masses) 4.62 D BH Distance to GC/BH (kpc) 8.46 TABLE 2 Average Velocity Measurements from Do et al. (2013) Variable Definition Value v x Central Velocity in X-Direction (km/s) v y Central Velocity in Y-Direction (km/s) 9.40 v z Central Velocity in Z-Direction (km/s) Updated Galactic Center Dynamics ρ [r] = ( r r b ) γ (1 + ( r r b ) δ ) (γ α) δ (12) A new stellar density distribution was adapted from Do et al. (2013) for stars in the inner GC. I normalized this distribution using the estimated total mass of the GC within the inner 1 parsec Schödel et al. (2009) to get a working density for stars per cubic parsec. Along with a normalization constant, the number density includes a factor of 0.2, used by Alexander and Sternberg to separate the stars that emit in the K-Band, from the total number of the stars. K-emitting stars are the only stars of any use to us as they are the only stars we observe. n [r] = 0.2 A normalization constant ρ [r] (13) In addition to a new number density, an average three-dimensional stellar velocity was used from Do et al. (2013). v = (14) 4.2. Updated K-Luminosity Function In choosing a new luminosity function, one has to consider which telescope/era to be used. Figure 5 shows an updated KLF for the Keck AO era, taken from W. M. Keck Observatorys projected NGAO Facility, 2014 (MSI (2014)).

5 Calculating a Stellar-Supermassive Black Hole Microlensing Rate 5 Events per year TABLE 3 Summary of lensing rates for different telescopes and integration limits Keck: Pre-AO K Thresh: 17 K Mag: 20.2 Micro. Lim: 45 pc Keck: AO K Thresh: 18.5 K Mag: Micro. Lim: 7 pc Keck: NGAO K Thresh: 22 K Mag: 19.7 Micro. Lim: 6.5 pc TMT K Thresh: 23 K Mag: Micro. Lim: 0.72 pc Inner pc Micro Limit 10 pc pc TMT K Thresh: 24 K Mag: Micro. Lim: 0.72 pc The blue curve is the KLF for current AO and an average magnitude of 17 has been found along with a threshold magnitude of about 18.5 (taking into account KLF incompleteness, or where the KLF begins to drop off in the range of stars we cannot see). The red curve is the KLF for Next Generation Adaptive Optics, with an average magnitude of about 19.7 and threshold of 22 mag. Figure 6 depicts a KLF for the TMT era. This KLF was adapted from Yelda et al. (2013) with an average magnitude of about and threshold of mag. It is important to mention that the basic structure of Alexander s rate equation was maintained, consisting of the integral form, arrangement of parameters, and consideration of observational time dependence. Individual parameters were revised as has been discussed and placed back into the skeleton structure of the equation. 5. RESULTS AND DISCUSSION After replacing the parameters, the same calculations were run again: for Keck AO, Keck NGAO, and TMT. Care was made to calculate the individual angular resolutions of these telescopes and their corresponding limits where microlensing becomes resolved lensing. The angular resolution of Keck AO was considered to be 60 mas, while the same value for Keck NGAO and TMT was considered to be 55 mas and 18 mas, respectively. These values were calculated using the equation for the angular resolution of the circular aperture, θ = 1.22 λ D where the midpoint wavelength for the K-band is 2.2 micrometers (the aperture diameter for Keck is 10 m and the aperture diameter for TMT is, of course, 30 m). Table 3 above summarizes lensing rates in the limit of near-continuous observation for multiple telescopes and different values for distance from the Galactic center. Out of curiosity, I recalculated the lensing rate for the pre-ao era (the same magnitude limits Alexander had used), and I changed the velocity distribution, black hole parameters, and stellar number density. Interestingly enough, the rate is still only a couple of events per thousand years. Moving on to Keck AO and NGAO era, we can expect to see 1-2 events every 5 years (if we include stars dwelling outside the inner few pc beyond the SMBH). Moving on to TMT era, we can practically expect to see one event per year extending outside the inner few pc behind the SMBH. It appears that our updated knowledge of the GC and occurring lensing events isn t what is bringing up the rate; it is the improved observational limits and instruments. With TMT, we can even expect to see multiple resolved events, as the microlensing limit lies in the inner 1 pc but the lensing rate continues to increase out to the inner 10 pc. For all instruments outside of the Keck pre-ao limit, if we extend at least 10 pc behind the SMBH in our observations, we should expect a lensing rate two orders of magnitude higher than that calculated by Alexander and Sternberg. These results are very promising: it is likely we will be able to observe multiple events over the next few decades, especially with the advent of TMT. The rate was calculated for all instruments considering only the inner 0.5 pc beyond the SMBH as that is the radial range of the data of the UCLA Galactic Center Group. Unfortunately, for Keck AO, the rate is only about 1 event per 30 years. It

6 6 Lipartito et al is not likely we will see more than one event (if that) out of a dataset only spanning 20 years. Figure 7 depicts a few plots which illustrate the time-dependent nature of the lensing rates. The revised calculations for pre-ao Keck have a strong dependence on the time between observing runs, whereas the AO and NGAO calculated rates remain rather constant with the increase in time. It appears that the improved observing parameters allows for the detection of a higher proportion of lensing events that are longer in duration, which would weaken the time dependent nature of the lensing rate (missing out on lensing events does not matter so much if there are so many long ones that will still be detectable). Figure 8 depicts updated TMT lensing rates. A similar lack of time dependence is seen for TMT. Once again, extending out to at least tens of parsecs allows for a substantial lensing rate, about 1 per 4 years with a conservative limit of 23 magnitude; about 1 per year with a more liberal limit of 24 magnitude. Regardless, certainly an improvement on the original results of Alexander and Sternberg. 6. FUTURE PLANS Moving forward, there would be several ways to improve upon this calculation. I mentioned earlier that an average 3-D stellar velocity was used in the updated calculation, adapted from Do et al. (2013). As a future move, it would be more correct to include a dependency in the velocity distribution on distance from the GC, as a means to more accurately represent stars found beyond the inner GC. Another means for progress would be to account more adeptly for factors which would prevent us from seeing stars we should otherwise be able to detect (which are otherwise within the observational limits of the telescope). This would include adding in updated parameters for gas extinction along with a mechanism of taking into account source confusion. We cannot detect all the stars predicted to exist, and the issue of detecting a source star against the confusion of multiple background stars (especially because the region of space being observed is so far and crowded) is an important one to consider when computing an accurate rate. Finally, another step to start considering is how we are going to detect and register these lensing events, especially if, as our rates have shown, multiple events might be occurring over an observational period of several years. We need to consider in particular the time duration of lensing events and whether some might be overlapping or happening at once. For microlensing, we need to consider how to differentiate these lensing events from each other and from the regular brightness fluctuations of the black hole. For resolved lensing, we need to consider how we are going to detect the lensed pairs, especially considering source confusion. As NGAO and the TMT become realities, these are questions we need to consider in order to make microlensing and lensing detection a valid observational endeavor. 7. CONCLUSIONS AND SIGNIFICANCE We have thus discovered that detection of microlensing and resolved lensing events from the supermassive black hole at the Galactic center is a viable possibility for the future generation of observational astronomy instruments: NGAO and TMT. Despite a rather low rate predicted by Alexander and Sternberg in 1999, we sought to update and redo their calculations with current parameters for the SMBH and the GC along with the detection limits of current and future telescopes. At the moment, we can expect to detect an event every few years with Keck AO and NGAO and perhaps about one event per year with the advent of TMT. Lensing events provide another means to observe the black hole, as we can only indirectly gain information on its nature by observing its interactions with objects around it. Observing the lensing by a black hole is yet another method of constraining parameters for its mass, distance from us, and the like. Bin-Nun (2010) suggests that the observing of microlensing events by the SMBH is a plausible method for learning about the black hole s space-time metric and constraining parameters for such a metric. Finally, while we have good confirmation of the workings of general relativity on a solar system and galactic scale (through other observations of lensing done by stars and galaxies), we have yet to confirm GR at the level of incredibly strong gravitational fields as found near a black hole. By observing a relativistic phenomenon done by the SMBH itself, we should be able to use lensing as a useful probe of GR at the SMBH level: comparing the theoretical predictions of relativistic lensing against obser-

7 Calculating a Stellar-Supermassive Black Hole Microlensing Rate 7 vational evidence. This is also an opportunity to discover where GR might break down and consider alternate theories of gravity (including quantum gravity). It appears that lensing and microlensing have the potential to take gravitational physics and GC astronomy into a new era of discovery of black hole and general relativistic properties Acknowledgements IL would like to thank foremost Dr. Andrea Ghez and the UCLA Galactic Center Group for their mentorship and guidance. IL is grateful to UCLA Physics and Astronomy as well and the NSF for their offer of this research position and granting of funding and resources. Special thanks to Samantha Chappell, Breann Sitarski, Leo Meyer, and Gunther Witzel for their assistance in this project. 2014, W. M. Keck Observatorys Next Generation Adaptive Optics Facility. March 12, 2014, MSIP 2014 Proposal Alexander, T. & Sternberg, A. 1999, ApJ, 520, 137 Bin-Nun, A. Y. 2010, Phys. Rev. D, 82, Do, T., Martinez, G. D., Yelda, S., Ghez, A., Bullock, J., Kaplinghat, M., Lu, J. R., Peter, A. H. G., & Phifer, K. 2013, ApJ, 779, L6 REFERENCES Ghez, A. M., Salim, S., Weinberg, N. N., Lu, J. R., Do, T., Dunn, J. K., Matthews, K., Morris, M. R., Yelda, S., Becklin, E. E., Kremenek, T., Milosavljevic, M., & Naiman, J. 2008, ApJ, 689, 1044 Schödel, R., Merritt, D., & Eckart, A. 2009, A&A, 502, 91 Yelda, S., Meyer, L., Ghez, A., & Do, T. 2013, in Proceedings of the Third AO4ELT Conference, ed. S. Esposito & L. Fini 8. FIGURES

8 8 Lipartito et al Fig. 1. The KLF generated by Alexander and Sternberg. An expected K magnitude, K of 20.2 was found by normalizing this distribution and finding the average magnitude.

9 Calculating a Stellar-Supermassive Black Hole Microlensing Rate 9 Fig. 2. Time-dependent microlensing rates for different threshold magnitudes (ranging from 16 to 19), generated from reproducing the calculations of Alexander and Sternberg. Smaller dots are rates I produced and larger dots are the rates from Alexander and Sternberg. Agreement is typically within an order of magnitude and is fairly consistent. Main result of 1-3 microlensing events per year in the limit of nearcontinuous observation is reproduced.

10 10 Lipartito et al Fig. 3. Time-dependent resolved lensing rates for different threshold magnitudes (ranging from 16 to 19), generated from reproducing the calculations of Alexander and Sternberg. Smaller dots are rates I produced and larger dots are the rates from Alexander. Agreement is typically within an order of magnitude and is fairly consistent. Resolved lensing rates are an order of magnitude smaller than microlensing rates for all magnitude thresholds.

11 Calculating a Stellar-Supermassive Black Hole Microlensing Rate 11 Fig. 4. Time-dependent total rates (microlensing and resolved lensing) for different threshold magnitudes (ranging from 16 to 19), generated from reproducing the calculations of Alexander and Sternberg. Smaller dots are rates I produced and larger dots are the rates from Alexander and Sternberg. Agreement is typically within an order of magnitude and is fairly consistent. Microlensing events dominate the total lensing rate as can be seen by comparing the different rates.

12 12 Lipartito et al Fig. 5. The KLF for AO era Keck (blue) and NGAO era Keck (red). An expected K AO magnitude, K of 18.5 was found by normalizing the AO distribution and finding the average magnitude. An expected K NGAO magnitude, K of 19.7 was found by normalizing the NGAO distribution and finding the average magnitude.

13 Calculating a Stellar-Supermassive Black Hole Microlensing Rate 13 Fig. 6. The expected KLF for TMT era observations. An expected K magnitude, K of was found by normalizing this distribution and finding the average magnitude.

14 14 Lipartito et al Fig. 7. Time-dependent total lensing rates for different Keck eras, up to 50 pc. These were generated from updating the calculations of Alexander and Sternberg. Lensing rates go up several orders of magnitude as we move into Keck AO and NGAO eras Fig. 8. Time-dependent lensing rates (including both separate microlensing and resolved lensing rates) for different TMT magnitude limit estimates. These were generated from updating the calculations of Alexander and Sternberg. Lensing rates for both magnitude limits are optimal if we can observe extending out to include stars at least 10 pc from the GC. With the upper magnitude limit of 24, we can expect to nearly see 1 event per year.